Exact sequence 'counterexample'

I would like to show that if the sequence:

is exact, then G need not necessarily be isomorphic to . Just one example of G that is not isomorphic to this (where the above sequence is still exact) would be enough. I've tried many and I can't seem to find a single valid one.. any hints?

This is not my area of expertise, so excuse me if I make a dumb error, but can't we just let G be the integers with the first map multiplication by 2, and the second map sending n to n(mod 2)?

Edit: I didn't check if is isomorphic to the direct sum. The obvious map is probably an isomorphism.

Originally Posted by Defunkt

If I'm not mistaken, you can take and define , by

then is injective, is surjective and , so the sequence is exact.

Edit: Ah, Steve beat me to it. Woops.

Yes, that is exactly the problem! Virtually any decent map you can come up with gives and this is isomorphic to . So really you need to be able to find something quite strange, like an exact sequence where . But I've tried many functions and it seems all but impossible.

Wait a second, is not isomorphic to . The latter group has torsion elements, like . The former group is torsion-free. (You could also invoke the Fundamental Theorem of Finitely Generated Abelian Groups.)

Wait a second, is not isomorphic to . The latter group has torsion elements, like . The former group is torsion-free. (You could also invoke the Fundamental Theorem of Finitely Generated Abelian Groups.)

Great! Don't know how I missed that.. the 'obvious' homomorphism between them looked like an isomorphism to me.

Great! Don't know how I missed that.. the 'obvious' homomorphism between them looked like an isomorphism to me.

You might need to check the difference between a short exact sequence and a split exact sequence. Being a short exact sequence is a necessary condition for being a split exact sequence. Given a short exact sequence, there are some additional conditions to be satisfied for being a split exact sequence.